Slightly off-topic here, but the second half of my signature, labelling is rarely important, is not totally irrelevant here, I think. Until this thread started, I would have answered "adverb", if anybody had asked me out of the blue what part of speech 'well' was in "I am well".
Now that I have thought about it, and realised that all other words that go in that slot are traditionally regarded as adjectives, I have decided that 'well' must be an adjective here. The dictionaries I have consulted agree.
But - does it matter what we call it?
Learners need to know that the normal response to the enquiry in BrE has been, and still is, for many of us, "I am well/fine/OK/depressed/etc". I am good", which appears to have been acceptable in AmE for a long time, is being used more and more in BrE. Whether these words should be labbelled as adverbs or adjectives is not important, in my opinion.